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Asymptotics for the higher-order derivative nonlinear Schrödinger equation

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The work of P. I. N. is partially supported by CONACYT project 283698 and PAPIIT project IN103221
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  • We study the Cauchy problem for the derivative higher-order nonlinear Schrödinger equation

    $ \begin{cases} i\partial_{t}v+\dfrac{a}{2}\partial_{x}^{2}v-\dfrac{b}{4}\partial_{x} ^{4}v = \left( \overline{\partial_{x}v}\right) ^{2},\text{ }t>1,\text{ } x\in\mathbb{R},\\ v\left( 1,x\right) = v_{0}\left( x\right) ,\text{ }x\in\mathbb{R}\text{,} \end{cases} $

    where $ a,b>0. $ Our aim is to prove global existence and calculate the large time asymptotics of solutions. We develop the factorization techniques originated in papers [13,10,12]. Also we follow the method of papers [9,11] to transform the quadratic nonlinearity to critical cubic nonlinearities similarly to the normal forms of Shatah [18].

    Mathematics Subject Classification: Primary: 35B40; Secondary: 35Q35.


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    [10] N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.  doi: 10.1007/s00033-007-7008-8.
    [11] N. Hayashi and P. I. Naumkin, Asymptotic behavior for a quadratic nonlinear Schrödinger equation, Electron. J. Differential Equations, 15 2008, 38 pp.
    [12] N. Hayashi and P.I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 25 pp. doi: 10.1063/1.4929657.
    [13] N. Hayashi and T. Ozawa, Scattering theory in the weighted $L^{2}(R^{n})$ spaces for some Sc rödinger equations, Ann. I. H. P. (Phys. Théor.), 48 (1988), 17-37.
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    [18] J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., 38 (1985), 685-696.  doi: 10.1002/cpa.3160380516.
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